If you change the domain name URL on the blog then you have to change the URL on the social networking sites as well.
All that's changing is the URL and everyone clicking the .wordpress.com URLs for either older content or new content will be seamlessly redirected to the same content under the new URLs as that is what domain mapping does. It will take 3 - 4 months for the content to re-indexed and that means the site will be unranked temporarily. After that is done the search positioning and page rank will usually be the same as it was previously. Read more here > http://onecoolsitebloggingtips.com/2012/04/02/do-i-need-a-domain/
In order to map to a domain one must have an underlying .wordpress.com subdomain blog to map from. If you have registered a .wordpress.com blog you log in under the exact same username account that registered the blog and purchase a domain name by purchasing an annually renewable domain mapping upgrade for the blog. http://en.support.wordpress.com/domains/
Dashboard > Store > My Upgrades is where you do the primary blog set up knowing it can take between 24 - 72 hours for domain name propagation to take place throughout the internet. You can view the DNS changes here > http://www.whatsmydns.net/ What's important during that stage is to be patient.